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Although the volume fraction is a constant for a statistically homogeneous random medium, on a spatially local level it fluctuates. We study the full distribution of volume fraction within an observation window of finite size for models of random media. A formula due to Lu and Torquato for the standard deviation or “coarseness’’ associated with the local volume fraction ξ is extended for the nth moment of ξ for any n. The distribution function FL of the local volume fraction of five different model microstructures is evaluated using analytical and computer-simulation methods for a wide range of window sizes and overall volume fractions. On the line, we examine a system of fully penetrable rods and a system of totally impenetrable rods formed by random sequential addition (RSA). In the plane, we study RSA totally impenetrable disks and fully penetrable aligned squares. In three dimensions, we study fully penetrable aligned cubes. In the case of fully penetrable rods, we will also simplify and numerically i...

An optimal design problem for piezoelectric composite hydrophones is considered. The hydrophone consists of parallel piezoelectric rods embedded in a poroustransversely isotropic polymer matrix. We find the shape, volume fraction, and spatial arrangement of the piezoceramic rods, and the structure of the matrix material that maximizes the hydrophone performance characteristics. We found that the optimal composite consists of a hexagonal array of rods with small volume fraction, in a highly anisotropic matrix that is characterized by negative Poisson's ratios in certain directions. The performance characteristics of hydrophones with such a matrix are significantly higher than those with anisotropic polymer matrix. The results can be viewed as theoretical upper bounds on the hydrophone performance.

We propose a model for statistically inhomogeneous two-phase random media, including functionally graded materials, consisting of inhomogeneous fully penetrable (Poisson distributed) spheres. This model can be constructed for any specified variation of volume fraction and permits one to represent and evaluate certain n-point correlation functions that statistically characterize the microstructure for this model. Unlike the case of statistically homogeneous media, the microstructure functions depend upon the absolute positions of their arguments. However, as with homogeneous random media, this microstructural information will be useful in obtaining rigorous estimates of the effective properties of such systems.

To what extent can the structure of a disordered heterogeneous material be reconstructed using limited but essentially exact structural information about the original system? We formulate a methodology based on simulated annealing to reconstruct both equilibrium and non-equilibrium dispersions of particles based only on correlation functions which statistically characterize the system. To test this method, we reconstruct dispersions from the radial distribution function (RDF) associated with the original system. Other statistical correlation functions are evaluated to compare how well the reconstructed system matches the original system. We show that for low-density systems or high-density systems with little particle aggregation, our reconstruction from the RDF reproduces the system fairly well. However, for dense systems with extensive clustering, RDF information is somewhat inadequate in being able to reconstruct the original system. We also show that one can produce a system with an RDF that is similar to the reference system, but with appreciably different structure. Finally, system-size effects are analyzed analytically.

We consider the problem of analytically determining the effective thermal conductivity of a composite material consisting of periodic arrays of spheres with interfacial resistance. We applied Rayle...

We generate jammed disordered packings of $100\ensuremath{\le}N\ensuremath{\le}2000$ monodisperse hard spheres in three dimensions whose strictly jammed backbones are demonstrated to be exactly isostatic with unprecedented numerical accuracy. This is accomplished by using the Torquato-Jiao (TJ) packing algorithm as a means of studying the maximally random jammed (MRJ) state. The rattler fraction of these packings converges towards $0.015$ in the infinite-system limit, which is markedly lower than previous estimates for the MRJ state using the Lubachevsky-Stillinger protocol. This is because the packings that the TJ algorithm creates are closer to the true MRJ state, as shown using bond-orientational and translational order metrics. The rattler pair correlation statistics exhibit strongly correlated behavior contrary to the conventional understanding that they be randomly (Poisson) distributed. Dynamically interacting ``polyrattlers'' may be found imprisoned in shared cages as well as interacting through ``bottlenecks'' in the backbone and these clusters are mainly responsible for the sharp increase in the rattler pair correlation function near contact. We discover the surprising existence of polyrattlers with cluster sizes of up to five rattlers (which is expected to increase with system size) and present a distribution of polyrattler occurrence as a function of cluster size and system size. We also enumerate all of the rattler interaction topologies we observe and present images of several examples, showing that MRJ packings of monodisperse spheres can contain large rattler cages while still obeying the strict jamming criterion. The backbone spheres that encage the rattlers are significantly hypostatic, implying that correspondingly hyperstatic regions must exist elsewhere in these isostatic packings. We also observe that rattlers in hard-sphere packings share an apparent connection with the low-temperature two-level system anomalies that appear in real amorphous insulators and semiconductors.

In the first paper of this series (DiStasio, Jr., Marcotte, Car, Stillinger, and Torquato, Phys. Rev. B 88, 134104 (2013)), we applied inverse statistical-mechanical techniques to study the extent to which targeted spin configurations on the square lattice can be ground states of finite-ranged radial spin-spin interactions. In this sequel, we enumerate all of the spin configurations within a unit cell on the one-dimensional integer lattice and the two-dimensional square lattice up to some modest size under periodic boundary conditions. We then classify these spin configurations into those that can or cannot be unique classical ground states of the aforementioned radial pair spin interactions and found the relative occurrences of these ground-state solution classes for different system sizes. As a result, we also determined the minimal radial extent of the spin-spin interaction potentials required to stabilize those configurations whose ground states are either unique or degenerate (i.e., those sharing the same radial spin-spin correlation function). This enumeration study has established that unique ground states are not limited to simple target configurations. However, we also found that many simple target spin configurations cannot be unique ground states.

The study of the packing of hard hyperspheres in $d$-dimensional Euclidean space ${\mathbb{R}}^{d}$ has been a topic of great interest in statistical mechanics and condensed matter theory. While the densest known packings are ordered in sufficiently low dimensions, it has been suggested that in sufficiently large dimensions, the densest packings might be disordered. The random sequential addition (RSA) time-dependent packing process, in which congruent hard hyperspheres are randomly and sequentially placed into a system without interparticle overlap, is a useful packing model to study disorder in high dimensions. Of particular interest is the infinite-time saturation limit in which the available space for another sphere tends to zero. However, the associated saturation density has been determined in all previous investigations by extrapolating the density results for nearly saturated configurations to the saturation limit, which necessarily introduces numerical uncertainties. We have refined an algorithm devised by us [S. Torquato, O. U. Uche, and F. H. Stillinger, Phys. Rev. E 74, 061308 (2006)] to generate RSA packings of identical hyperspheres. The improved algorithm produce such packings that are guaranteed to contain no available space in a large simulation box using finite computational time with heretofore unattained precision and across the widest range of dimensions ($2\ensuremath{\le}d\ensuremath{\le}8$). We have also calculated the packing and covering densities, pair correlation function ${g}_{2}(r)$, and structure factor $S(k)$ of the saturated RSA configurations. As the space dimension increases, we find that pair correlations markedly diminish, consistent with a recently proposed ``decorrelation'' principle, and the degree of ``hyperuniformity'' (suppression of infinite-wavelength density fluctuations) increases. We have also calculated the void exclusion probability in order to compute the so-called quantizer error of the RSA packings, which is related to the second moment of inertia of the average Voronoi cell. Our algorithm is easily generalizable to generate saturated RSA packings of nonspherical particles.

We report the first experimental demonstration of a TE-polarization photonic band gap (PBG) in a 2D isotropic hyperuniform disordered solid (HUDS) made of dielectric media with a dielectric index contrast of 1.6:1, very low for PBG formation. The solid is composed of a connected network of dielectric walls enclosing air-filled cells. Direct comparison with photonic crystals and quasicrystals permitted us to investigate band-gap properties as a function of increasing rotational isotropy. We present results from numerical simulations proving that the PBG observed experimentally for HUDS at low index contrast has zero density of states. The PBG is associated with the energy difference between complementary resonant modes above and below the gap, with the field predominantly concentrated in the air or in the dielectric. The intrinsic isotropy of HUDS may offer unprecedented flexibilities and freedom in applications (i. e. defect architecture design) not limited by crystalline symmetries.

Inverse statistical-mechanical methods have recently been employed to design optimized short-range radial (isotropic) pair potentials that robustly produce novel targeted classical ground-state many-particle configurations. The target structures considered in those studies were low-coordinated crystals with a high degree of symmetry. In this paper, we further test the fundamental limitations of radial pair potentials by targeting crystal structures with appreciably less symmetry, including those in which the particles have different local structural environments. These challenging target configurations demanded that we modify previous inverse optimization techniques. In particular, we first find local minima of a candidate enthalpy surface and determine the enthalpy difference ΔH between such inherent structures and the target structure. Then we determine the lowest positive eigenvalue λ(0) of the Hessian matrix of the enthalpy surface at the target configuration. Finally, we maximize λ(0)ΔH so that the target structure is both locally stable and globally stable with respect to the inherent structures. Using this modified optimization technique, we have designed short-range radial pair potentials that stabilize the two-dimensional kagome crystal, the rectangular kagome crystal, and rectangular lattices, as well as the three-dimensional structure of the CaF(2) crystal inhabited by a single-particle species. We verify our results by cooling liquid configurations to absolute zero temperature via simulated annealing and ensuring that such states have stable phonon spectra. Except for the rectangular kagome structure, all of the target structures can be stabilized with monotonic repulsive potentials. Our work demonstrates that single-component systems with short-range radial pair potentials can counterintuitively self-assemble into crystal ground states with low symmetry and different local structural environments. Finally, we present general principles that offer guidance in determining whether certain target structures can be achieved as ground states by radial pair potentials.

We report the results of highly sensitive transmission X-ray scattering measurements performed at the Advanced Photon Source, Argonne National Laboratory, on nearly fully dense high-purity amorphous-silicon (a-Si) samples for the purpose of determining their degree of hyperuniformity. A perfectly hyperuniform structure has complete suppression of infinite-wavelength density fluctuations, or, equivalently, the structure factor S ( q →0) = 0; the smaller the value of S (0), the higher the degree of hyperuniformity. Annealing was observed to increase the degree of hyperuniformity in a-Si where we found S (0) = 0.0075 (±0.0005), which is significantly below the computationally determined lower bound recently suggested by de Graff and Thorpe [de Graff AMR, Thorpe MF (2010) Acta Crystallogr A 66(Pt 1):22–31] based on studies of continuous random network models, but consistent with the recently proposed nearly hyperuniform network picture of a-Si. Increasing hyperuniformity is correlated with narrowing of the first diffraction peak and extension of the range of oscillations in the pair distribution function.

We introduce the concept of nearly hyperuniform network (NHN) structures as alternatives to the conventional continuous random network (CRN) models for amorphous tetrahedrally coordinated solids, such as amorphous silicon (a-Si). A hyperuniform solid has a structure factor $S(k)$ that approaches zero as the wavenumber $k\ensuremath{\rightarrow}0$. We define a NHN as an amorphous network whose structure factor $S(k\ensuremath{\rightarrow}0)$ is smaller than the liquid value at the melting temperature. Using a novel implementation of the Stillinger-Weber potential for the interatomic interactions, we show that the energy landscape for a spectrum of NHNs includes a sequence of local minima with an increasing degree of hyperuniformity [smaller $S(k\ensuremath{\rightarrow}0)$] that is significantly below the frozen-liquid value and that correlates with other measurable features in $S(k)$ at intermediate and large $k$ and with the width of the electronic band gap.

Finding the densest sphere packing in $d$-dimensional Euclidean space $\mathbb{R}^d$ is an outstanding fundamental problem with relevance in many fields, including the ground states of molecular systems, colloidal crystal structures, coding theory, discrete geometry, number theory, and biological systems. Numerically generating the densest sphere packings becomes very challenging in high dimensions due to an exponentially increasing number of possible sphere contacts and sphere configurations, even for the restricted problem of finding the densest lattice sphere packings. In this paper, we apply the Torquato-Jiao packing algorithm, which is a method based on solving a sequence of linear programs, to robustly reproduce the densest known lattice sphere packings for dimensions 2 through 19. We show that the TJ algorithm is appreciably more efficient at solving these problems than previously published methods. Indeed, in some dimensions, the former procedure can be as much as three orders of magnitude faster at finding the optimal solutions than earlier ones. We also study the suboptimal local density-maxima solutions (inherent structures or "extreme" lattices) to gain insight about the nature of the topography of the "density" landscape., Comment: 23 pages, 3 figures

Previous attempts to simulate disordered binary sphere packings have been limited in producing mechanically stable, isostatic packings across a broad spectrum of packing fractions. Here we report that disordered strictly jammed binary packings (packings that remain mechanically stable under general shear deformations and compressions) can be produced with an anomalously large range of average packing fractions $0.634\ensuremath{\le}\ensuremath{\phi}\ensuremath{\le}0.829$ for small to large sphere radius ratios $\ensuremath{\alpha}$ restricted to $\ensuremath{\alpha}\ensuremath{\ge}0.100$. Surprisingly, this range of average packing fractions is obtained for packings containing a subset of spheres (called the backbone) that are exactly strictly jammed, exactly isostatic, and also generated from random initial conditions. Additionally, the average packing fractions of these packings at certain $\ensuremath{\alpha}$ and small sphere relative number concentrations $x$ approach those of the corresponding densest known ordered packings. These findings suggest for entropic reasons that these high-density disordered packings should be good glass formers and that they may be easy to prepare experimentally. We also identify an unusual feature of the packing fraction of jammed backbones (packings with rattlers excluded). The backbone packing fraction is about $0.624$ over the majority of the $\ensuremath{\alpha}$-$x$ plane, even when large numbers of small spheres are present in the backbone. Over the (relatively small) area of the $\ensuremath{\alpha}$-$x$ plane where the backbone is not roughly constant, we find that backbone packing fractions range from about $0.606$ to $0.829$, with the volume of rattler spheres comprising between $1.6%$ and $26.9%$ of total sphere volume. To generate isostatic strictly jammed packings, we use an implementation of the Torquato-Jiao sequential linear programming algorithm [Phys. Rev. E 82, 061302 (2010)], which is an efficient producer of inherent structures (mechanically stable configurations at the local maxima in the density landscape). The identification and explicit construction of binary packings with such high packing fractions could have important practical implications for granular composites where density is critical both to material properties and fabrication cost, including for solid propellants, concrete, and ceramics. The densities and structures of jammed binary packings at various $\ensuremath{\alpha}$ and $x$ are also relevant to the formation of a glass phase in multicomponent metallic systems.

Using finite-difference time domain and band structure computer simulations, we show that it is possible to construct optical cavities and waveguide architectures in hyperuniform disordered photonic solids that are unattainable in photonic crystals. The cavity modes can be classified according to the symmetry (monopole, dipole, quadrupole, etc.) of the confined electromagnetic wave pattern. Owing to the isotropy of the band-gap characteristics of hyperuniform disordered solids, high-quality waveguides with free-form geometries (e.g., arbitrary bending angles) can be constructed that are unprecedented in periodic or quasiperiodic solids. These capabilities have implications for many photonic applications.

We show analytically that the $[0,1]$, $[1,1]$ and $[2,1]$ Pad{\'e} approximants of the mean cluster number $S(p)$ for site and bond percolation on general $d$-dimensional lattices are upper bounds on this quantity in any Euclidean dimension $d$, where $p$ is the occupation probability. These results lead to certain lower bounds on the percolation threshold $p_c$ that become progressively tighter as $d$ increases and asymptotically exact as $d$ becomes large. These lower-bound estimates depend on the structure of the $d$-dimensional lattice and whether site or bond percolation is being considered. We obtain explicit bounds on $p_c$ for both site and bond percolation on five different lattices: $d$-dimensional generalizations of the simple-cubic, body-centered-cubic and face-centered-cubic Bravais lattices as well as the $d$-dimensional generalizations of the diamond and kagom{\'e} (or pyrochlore) non-Bravais lattices. These analytical estimates are used to assess available simulation results across dimensions (up through $d=13$ in some cases). It is noteworthy that the tightest lower bound provides reasonable estimates of $p_c$ in relatively low dimensions and becomes increasingly accurate as $d$ grows. We also derive high-dimensional asymptotic expansions for $p_c$ for the ten percolation problems and compare them to the Bethe-lattice approximation. Finally, we remark on the radius of convergence of the series expansion of $S$ in powers of $p$ as the dimension grows., Comment: 37 pages, 5 figures

We apply inverse statistical-mechanical methods to find a simple family of optimized isotropic, monotonic pair potentials (that may be experimentally realizable) whose classical ground state is the diamond crystal for the widest possible pressure range, subject to certain constraints (e.g., desirable phonon spectra). We also ascertain the ground-state phase diagram for a specific optimized potential to show that other crystal structures arise for pressures outside the diamond stability range. Cooling disordered configurations interacting with our optimized potential to absolute zero frequently leads to the desired diamond crystal ground state, revealing that the capture basin for the global energy minimum is large and broad relative to the local energy minima basins.

A set of lower bounds on the continuum percolation threshold $\eta_c$ of overlapping convex hyperparticles of general nonspherical (anisotropic) shape with a specified orientational probability distribution in $d$-dimensional Euclidean space have been derived [S. Torquato, J. Chem. Phys. {\bf 136}, 054106 (2012)]. The simplest of these lower bounds is given by $\eta_c \ge v/v_{ex}$, where $v_{ex}$ is the $d$-dimensional exclusion volume of a hyperparticle and $v$ is its $d$-dimensional volume. In order to study the effect of dimensionality on the threshold $\eta_c$ of overlapping nonspherical convex hyperparticles with random orientations here, we obtain a scaling relation for $\eta_c$ that is based on this lower bound and a conjecture that hyperspheres provide the highest threshold among all convex hyperparticle shapes for any $d$. This scaling relation exploits the principle that low-dimensional continuum percolation behavior encodes high-dimensional information. We derive a formula for the exclusion volume $v_{ex}$ of a hyperparticle in terms of its $d$-dimensional volume $v$, surface area $s$ and {\it radius of mean curvature} ${\bar R}$ (or, equivalently, {\it mean width}). These basic geometrical properties are computed for a wide variety of nonspherical hyperparticle shapes with random orientations across all dimensions, including, among other shapes, various polygons for $d=2$, Platonic solids, spherocylinders, parallepipeds and zero-volume plates for $d=3$ and their appropriate generalizations for $d \ge 4$. We then compute the lower bound and scaling relation for $\eta_c$ for this comprehensive set of continuum percolation models across dimensions. We show that the scaling relation provides accurate {\it upper-bound} estimates of the threshold $\eta_c$ across dimensions and becomes increasingly accurate as the space $d$ increases., Comment: 37 pages, 5 figures, 9 tables, to appear in Phys. Rev. E

Recently, disordered photonic media and random textured surfaces have attracted increasing attention as strong light diffusers with broadband and wide-angle properties. We report the experimental realization of an isotropic complete photonic band gap (PBG) in a 2D disordered dielectric structure. This structure is designed by a constrained optimization method, which combines advantages of both isotropy due to disorder and controlled scattering properties due to low-density fluctuations (hyperuniformity) and uniform local topology. Our experiments use a modular design composed of Al 2 O 3 walls and cylinders arranged in a hyperuniform disordered network. We observe a complete PBG in the microwave region, in good agreement with theoretical simulations, and show that the intrinsic isotropy of this unique class of PBG materials enables remarkable design freedom, including the realization of waveguides with arbitrary bending angles impossible in photonic crystals. This experimental verification of a complete PBG and realization of functional defects in this unique class of materials demonstrate their potential as building blocks for precise manipulation of photons in planar optical microcircuits and has implications for disordered acoustic and electronic band gap materials.

We present comprehensive results of large‐scale molecular dynamics and Monte Carlo simulations of systems of dense hard spheres at volume fraction φ along the disordered, metastable branch of the phase diagram from the freezing‐point φf to random close packing volume φc. It is shown that many previous simulations contained deficiencies caused by crystallization and finite‐size effects. We quantify the degree of local crystallization through an order parameter and study it as a function of time and initial conditions to determine the necessary conditions to obtain truly random systems. This ordering criterion is used to show that previous methods employed to ascertain the degree of randomness are inadequate. A careful study of the pressure is also carried out along the entire metastable branch. In the vicinity of the random‐close packing fraction, we show that the pressure scales as (φc−φ)−γ, where γ=1 and φc=0.644±0.005. Contrary to previous studies, we find no evidence of a thermodynamic glass transition.

Various clustering properties of d-dimensional overlapping (i.e., Poisson distributed) spheres are investigated. We evaluate ${\mathit{n}}_{\mathit{k}}$, the average number of connected clusters of k particles (called k-mers) per unit particle, for k=2,3,4 and ${\mathit{v}}_{\mathit{k}}$, the expected volume of a k-mer, for k=2,3,4 by using our general expressions for these quantities for d=1, 2, or 3. We use these calculations to obtain low-density expansions of various averaged cluster numbers and volumes, which can be obtained from the ${\mathit{n}}_{\mathit{k}}$ and ${\mathit{v}}_{\mathit{k}}$. We study the behavior of these cluster statistics as the percolation threshold is approached from below, and we rigorously show that two of these averaged quantities do not diverge for d\ensuremath{\geqslant}2. \textcopyright{} 1996 The American Physical Society.

We present a study of the morphology and bulk physical properties of a Fontainebleau sandstone via an X ray tomographic analysis. Synchrotron-based X ray tomographic techniques provide us with a high-resolution (7.5 μm), three-dimensional digitized representation of the sandstone that leaves the sample intact and unaltered. To estimate a wide spectrum of bulk properties of the Fontainebleau sandstone specimen, we extract from this image a number of different correlation functions that statistically characterize the pore-space morphology and relevant pore-space length and time scales. These statistical measures are obtainable from lineal, plane, and/or volume measurements and include the porosity, specific surface, two-point and three-point probability functions, lineal-path function, chord-length distribution function, pore-size distribution function, and coarseness. The pore-size distribution function, in particular, contains a certain level of connectedness information and accordingly can only be obtained from a three-dimensional representation of the sample. Many bulk properties of the sandstone, such as the mean survival time τ (obtainable from Nuclear Magnetic Resonance relaxation studies), fluid permeability k, effective electrical and thermal conductivities, and effective elastic moduli, can be estimated using the aforementioned statistical correlation functions. Specifically, the electrical conductivity (or, equivalently, the formation factor F), mean survival time, and fluid permeability are determined using rigorous bounds. The mean survival time and fluid permeability are also found using direct simulation techniques and cross-property relations, respectively. One such cross-property relation for k depending on τ and F gives a permeability estimate that is within a factor of 2 of the experimental result.

Certain hierarchical laminates with a wide separation of length scales are known theoretically to have optimal transport and mechanical properties. We derive analytical expressions for the {ital n}-point probability functions that statistically characterize the microstructure for more general hierarchical laminates with an arbitrary number of stages and a finite separation of length scales. Using two-point probability information, we rigorously bound the effective conductivity (or dielectric constant) tensor for macroscopically anisotropic laminates and study how the separation of length scales affects the effective properties. {copyright} {ital 1996 The American Physical Society.}

For isotropic two-phase composites, we derive phase-interchange inequalities for the bulk and shear moduli in two and three space dimensions. We find optimal microstructures that realize part of the bull, and shear moduli bounds in two dimensions. Geometrical-parameter bounds and the translation method are used to prove the results. The phase-interchange relations are applied to composites with cavities or a perfectly rigid phase, composites near the percolation threshold, incompressible composites, composites with equal bulk or shear phase moduli, and symmetric composites.

Rigorous cross-property bounds that connect the effective electrical conductivity σ * and the effective bulk modulus K * of any isotropic two-phase composite are derived when the volume fractions of the phases are either specified or unknown. These bounds enclose lens-shaped regions in the σ * – K * plane, portions of which are attain­able by certain microgeometries and thus are optimal. Our cross-property bounds apply also to anisotropic composites with cubic symmetry. The bounds are applied to some general situations, as well as to specific microgeometries, including regular and random arrays of spheres and hierarchical geometries corresponding to effective-medium theories. It is shown that knowledge of the effective conductivity can yield sharp estimates of the effective bulk modulus (and vice versa), even in cases where there is a wide disparity in the phase properties.

We find bounds on the effective elastic moduli of cracked materials in terms of the effective conductivity of such media. These represent the first non-trivial bounds on the effective properties of cracked media which are independent of the shapes and spatial distribution of the cracks. Different approximations for the elastic moduli of cracked media are tested against our bounds. The microgeometries of cracks that satisfy the bounds exactly are identified.

The probability of finding a nearest neighbor at some radial distance from a reference point in many-particle systems is of fundamental importance in a host of fields in the physical as well as biological sciences. We have derived exact analytical expressions for nearest-neighbor probability functions for particles deposited on a line during a random sequential adsorption process for all densities, i.e., up to the jamming limit. Using these results, we find the mean nearest-neighbor distance {lambda} as a function of the packing fraction, and discuss it in light of recent theorems derived for general ergodic and isotropic packings of hard spheres. {copyright} {ital 1996 The American Physical Society.}

We study bounds on the effective conductivity and elastic moduli of two-phase isotropic composites that depend on geometrical parameters that take into account up to three-point statistical information concerning the composite microstructure. We summarize existing bounds, apply a special fractional linear transformation to simplify their functional forms, and describe two approaches to improve such bounds. These approaches allow us to get new and improved geometrical-parameter bounds on the elastic moduli of two-dimensional composites. Applications of the bounds for effective-medium geometries as well as random arrays of aligned fibers in a matrix are discussed.

We study statistical and structural properties of extreme lattices, which are the local minima in the density landscape of lattice sphere packings in d-dimensional Euclidean space R-d. Specifically, we ascertain statistics of the densities and kissing numbers as well as the numbers of distinct symmetries of the packings for dimensions 8 through 13 using the stochastic Voronoi algorithm. The extreme lattices in a fixed dimension of space d (d >= 8) are dominated by typical lattices that have similar packing properties, such as packing densities and kissing numbers, while the best and the worst packers are in the long tails of the distribution of the extreme lattices. We also study the validity of the recently proposed decorrelation principle, which has important implications for sphere packings in general. The degree to which extreme-lattice packings decorrelate as well as how decorrelation is related to the packing density and symmetry of the lattices as the space dimension increases is also investigated. We find that the extreme lattices decorrelate with increasing dimension, while the least symmetric lattices decorrelate faster.

We present an analysis of the effect of digitization at a moderate resolution on the determination of various morphological quantities for a model three‐dimensional digitized medium. Specifically, we study systems of digitized overlapping spheres which have many of the features found in man‐made and geologic materials. The goal of this paper is to demonstrate which quantities are most sensitive to the process of digitization and resolution. In addition, the question of whether three‐dimensional data sets are necessary to obtain reliable results concerning material structure is answered in the affirmative.

We derive rigorous three‐point upper and lower bounds on the effective bulk and shear moduli of a two‐phase material composed of equisized spheres randomly distributed throughout a matrix. Our approach is analogous to previously derived three‐point cluster bounds on the effective conductivity of suspensions of spheres. Our bounds on the effective elastic moduli are then compared to other known three‐point bounds for statistically homogeneous and isotropic random materials. For the case of totally impenetrable spheres, the bulk modulus bounds are shown to be equivalent to the Beran–Molyneux bounds, and the shear modulus bounds are compared to the McCoy and Milton–Phan‐Thien bounds. For the case of fully penetrable spheres, our bounds are shown to be simple analytical expressions, in contrast to the numerical quadratures required to evaluate the other three‐point bounds.

We derive the first nontrivial rigorous bounds on the mean distance between nearest neighbors $\ensuremath{\lambda}$ in ergodic, isotropic packings of hard $D$-dimensional spheres that depend on the packing fraction and nearest-neighbor distribution function. Several interesting implications of these bounds for equilibrium as well as nonequilibrium ensembles are explored. For an equilibrium ensemble, we find accurate analytical approximations for $\ensuremath{\lambda}$ for $D\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$ and 3 that apply up to random close packing. Our theoretical results are in excellent agreement with available computer-simulation data.

We present an efficient and fast simulation technique to determine the mean survival time τ of a Brownian particle diffusing among a digitized lattice‐based domain of traps. Following the first‐passage time ideas of Torquato and Kim [Appl. Phys. Lett. 55, 1847 (1989)], the algorithm is based on the known solution for the mean first passage time of a Brownian particle in a cube. The mean survival time, the inverse of the trapping rate, is obtained for a variety of configurations involving digitized spheres. Since the survival time is highly sensitive to the surface area and associated resolution, the results provide a means of determining the relation between the survival time of a real material and its digitized representation. In general, lower resolution images give rise to a diminished mean survival time and, thus, a lower bound on the true mean survival time τ. Digitization can affect other transport properties in which the interface plays a major role, e.g., the fluid permeability associated with flow in porous media. We demonstrate both analytically and computationally that the mean survival time for the digitized medium converges to the continuum value in the high‐resolution limit.

The two-point cluster functionC2(r1,r2) provides a measure of clustering in continuum models of disordered many-particle systems and thus is a useful signature of the microstructure. For a two-phase disordered medium,C2(r1,r2) is defined to be the probability of finding two points at positionsr1 andr2 in thesame cluster of one of the phases. An exact analytical expression is found for the two-point cluster functionC2(r1,r2) of a one-dimensional continuumpercolation model of Poisson-distributed rods (for an arbitrary number density) using renewal theory. We also give asymptotic formulas for the tail probabilities. Along the way we find exact results for other cluster statistics of this continuum percolation model, such as the cluster size distribution, mean number of clusters, and two-point blocking function.

The emergence of invasive and metastatic behavior in malignant tumors can often lead to fatal outcomes for patients. The collective malignant tumor behavior resulting from the complex tumor-host interactions and the interactions between the tumor cells is currently poorly understood. In this paper, we employ a cellular automaton (CA) model to investigate microenvironment-enhanced malignant behaviors and morphologies of in vitro avascular invasive solid tumors in three dimensions. Our CA model incorporates a variety of microscopic-scale tumor-host interactions, including the degradation of the extracellular matrix by the malignant cells, nutrient-driven cell migration, pressure buildup due to the deformation of the microenvironment by the growing tumor, and its effect on the local tumor-host interface stability. Moreover, the effects of cell-cell adhesion on tumor growth are explicitly taken into account. Specifically, we find that while strong cell-cell adhesion can suppress the invasive behavior of the tumors growing in soft microenvironments, cancer malignancy can be significantly enhanced by harsh microenvironmental conditions, such as exposure to high pressure levels. We infer from the simulation results a qualitative phase diagram that characterizes the expected malignant behavior of invasive solid tumors in terms of two competing malignancy effects: the rigidity of the microenvironment and cell-cell adhesion. This diagram exhibits phase transitions between noninvasive and invasive behaviors. We also discuss the implications of our results for the diagnosis, prognosis, and treatment of malignant tumors.

The small wavenumber $k$ behavior of the structure factor $S(k)$ of overcompressed amorphous hard-sphere configurations was previously studied for a wide range of densities up to the maximally random jammed state, which can be viewed as a prototypical glassy state [A. Hopkins, F. H. Stillinger and S. Torquato, Phys. Rev. E, 86, 021505 (2012)]. It was found that a precursor to the glassy jammed state was evident long before the jamming density was reached as measured by a growing nonequilibrium length scale extracted from the volume integral of the direct correlation function $c(r)$, which becomes long-ranged as the critical jammed state is reached. The present study extends that work by investigating via computer simulations two different atomic models: the single-component Z2 Dzugutov potential in three dimensions and the binary-mixture Kob-Andersen potential in two dimensions. Consistent with the aforementioned hard-sphere study, we demonstrate that for both models a signature of the glass transition is apparent well before the transition temperature is reached as measured by the length scale determined from from the volume integral of the direct correlation function in the single-component case and a generalized direct correlation function in the binary-mixture case. The latter quantity is obtained from a generalized Orstein-Zernike integral equation for a certain decoration of the atomic point configuration. We also show that these growing length scales, which are a consequence of the long-range nature of the direct correlation functions, are intrinsically nonequilibrium in nature as determined by an index $X$ that is a measure of deviation from thermal equilibrium. It is also demonstrated that this nonequilibrium index, which increases upon supercooling, is correlated with a characteristic relaxation time scale., 26 pages, 14 figures

In the first paper of this series [S. Torquato, J. Chem. Phys. {\bf 136}, 054106 (2012)], analytical results concerning the continuum percolation of overlapping hyperparticles in $d$-dimensional Euclidean space $\mathbb{R}^d$ were obtained, including lower bounds on the percolation threshold. In the present investigation, we provide additional analytical results for certain cluster statistics, such as the concentration of $k$-mers and related quantities, and obtain an upper bound on the percolation threshold $\eta_c$. We utilize the tightest lower bound obtained in the first paper to formulate an efficient simulation method, called the {\it rescaled-particle} algorithm, to estimate continuum percolation properties across many space dimensions with heretofore unattained accuracy. This simulation procedure is applied to compute the threshold $\eta_c$ and associated mean number of overlaps per particle ${\cal N}_c$ for both overlapping hyperspheres and oriented hypercubes for $ 3 \le d \le 11$. These simulations results are compared to corresponding upper and lower bounds on these percolation properties. We find that the bounds converge to one another as the space dimension increases, but the lower bound provides an excellent estimate of $\eta_c$ and ${\cal N}_c$, even for relatively low dimensions. We confirm a prediction of the first paper in this series that low-dimensional percolation properties encode high-dimensional information. We also show that the concentration of monomers dominate over concentration values for higher-order clusters (dimers, trimers, etc.) as the space dimension becomes large. Finally, we provide accurate analytical estimates of the pair connectedness function and blocking function at their contact values for any $d$ as a function of density., Comment: 24 pages, 10 figures

The problem of tiling or tessellating (i.e., completely filling) three-dimensional Euclidean space R(3) with polyhedra has fascinated people for centuries and continues to intrigue mathematicians and scientists today. Tilings are of fundamental importance to the understanding of the underlying structures for a wide range of systems in the biological, chemical, and physical sciences. In this paper, we enumerate and investigate the most comprehensive set of tilings of R(3) by any two regular polyhedra known to date. We find that among all of the Platonic solids, only the tetrahedron and octahedron can be combined to tile R(3). For tilings composed of only congruent tetrahedra and congruent octahedra, there seem to be only four distinct ratios between the sides of the two polyhedra. These four canonical periodic tilings are, respectively, associated with certain packings of tetrahedra (octahedra) in which the holes are octahedra (tetrahedra). Moreover, we derive two families of an uncountably infinite number of periodic tilings of tetrahedra and octahedra that continuously connect the aforementioned four canonical tilings with one another, containing the previously reported Conway-Jiao-Torquato family of tilings [Conway et al., Proc. Natl. Acad. Sci. USA 108, 11009 (2011)] as a special case. For tilings containing infinite planar surfaces, nonperiodic arrangements can be easily generated by arbitrary relative sliding along these surfaces. We also find that there are three distinct canonical periodic tilings of R(3) by congruent regular tetrahedra and congruent regular truncated tetrahedra associated with certain packings of tetrahedra (truncated tetrahedra) in which the holes are truncated tetrahedra (tetrahedra). Remarkably, we discover that most of the aforementioned periodic tilings can be obtained by "retessellating" the well-known tiling associated with the face-centered-cubic lattice, i.e., by combining the associated fundamental tiles (regular tetrahedra and octahedra) to form larger polyhedra.

A two-point correlation function provides a crucial yet an incomplete characterization of a microstructure because distinctly different microstructures may have the same correlation function. In an earlier Letter [Gommes, Jiao, and Torquato, Phys. Rev. Lett. 108, 080601 (2012)], we addressed the microstructural degeneracy question: What is the number of microstructures compatible with a specified correlation function? We computed this degeneracy, i.e., configurational entropy, in the framework of reconstruction methods, which enabled us to map the problem to the determination of ground-state degeneracies. Here, we provide a more comprehensive presentation of the methodology and analyses, as well as additional results. Since the configuration space of a reconstruction problem is a hypercube on which a Hamming distance is defined, we can calculate analytically the energy profile of any reconstruction problem, corresponding to the average energy of all microstructures at a given Hamming distance from a ground state. The steepness of the energy profile is a measure of the roughness of the energy landscape associated with the reconstruction problem, which can be used as a proxy for the ground-state degeneracy. The relationship between this roughness metric and the ground-state degeneracy is calibrated using a Monte Carlo algorithm for determining the ground-state degeneracy of a variety of microstructures, including realizations of hard disks and Poisson point processes at various densities as well as those with known degeneracies (e.g., single disks of various sizes and a particular crystalline microstructure). We show that our results can be expressed in terms of the information content of the two-point correlation functions. From this perspective, the a priori condition for a reconstruction to be accurate is that the information content, expressed in bits, should be comparable to the number of pixels in the unknown microstructure. We provide a formula to calculate the information content of any two-point correlation function, which makes our results broadly applicable to any field in which correlation functions are employed.

Maximally random jammed states of hard spheres are prototypical glasses. We study the small wavenumber $k$ behavior of the structure factor $S(k)$ of overcompressed million-sphere packings as a function of density up to the jammed state. We find both a precursor to the glassy jammed state evident long before the jamming density is reached and an associated growing length scale, extracted from the volume integral of the direct correlation function $c(r)$, which diverges at the "critical" jammed state. We also define a nonequilibrium index $X$ and use it to demonstrate that the packings studied are intrinsically nonequilibrium in nature well before the critical point is reached., Comment: 13 pages, 5 figures

Biopolymer networks are of fundamental importance to many biological processes in normal and tumorous tissues. In this paper, we employ the panoply of theoretical and simulation techniques developed for characterizing heterogeneous materials to quantify the microstructure and effective diffusive transport properties (diffusion coefficient $D_e$ and mean survival time $\tau$) of collagen type I networks at various collagen concentrations. In particular, we compute the pore-size probability density function $P(\delta)$ for the networks and present a variety of analytical estimates of the effective diffusion coefficient $D_e$ for finite-sized diffusing particles. The Hashin-Strikman upper bound on the effective diffusion coefficient $D_e$ and the pore-size lower bound on the mean survival time $\tau$ are used as benchmarks to test our analytical approximations and numerical results. Moreover, we generalize the efficient first-passage-time techniques for Brownian-motion simulations in suspensions of spheres to the case of fiber networks and compute the associated effective diffusion coefficient $D_e$ as well as the mean survival time $\tau$, which is related to nuclear magnetic resonance (NMR) relaxation times. Specifically, the Torquato approximation provides the most accurate estimates of $D_e$ for all collagen concentrations among all of the analytical approximations we consider. We formulate a universal curve for $\tau$ for the networks at different collagen concentrations. We apply rigorous cross-property relations to estimate the effective bulk modulus of collagen networks from a knowledge of the effective diffusion coefficient computed here., Comment: 23 pages, 9 figures

We have recently devised organizing principles to obtain maximally dense packings of the Platonic and Archimedean solids, and certain smoothly-shaped convex nonspherical particles [Torquato and Jiao, Phys. Rev. E 81, 041310 (2010)]. Here we generalize them in order to guide one to ascertain the densest packings of other convex nonspherical particles as well as concave shapes. Our generalized organizing principles are explicitly stated as four distinct propositions. We apply and test all of these organizing principles to the most comprehensive set of both convex and concave particle shapes to date, including Catalan solids, prisms, antiprisms, cylinders, dimers of spheres and various concave polyhedra. We demonstrate that all of the densest known packings associated with this wide spectrum of nonspherical particles are consistent with our propositions. Among other applications, our general organizing principles enable us to construct analytically the densest known packings of certain convex nonspherical particles, including spherocylinders, "lens-shaped" particles, square pyramids and rhombic pyramids. Moreover, we show how to apply these principles to infer the high-density equilibrium crystalline phases of hard convex and concave particles. We also discuss the unique packing attributes of maximally random jammed packings of nonspherical particles., Comment: 37 pages, 14 figures

Using a new class of isotropic disordered photonic bandgap material, we demonstrate freeform photonics waveguides with arbitrary shapes and multi-channel frequency splitters with versatile tuning abilities, useful for signal processing and all-optical circuit applications.

Results from the numerical simulation of Stokes flow through random packings of nonoverlapping or overlapping sphers and a [ital scaling] [ital ansatz] are used to obtain universal curves for the fluid permeability. The scaling ansatz is motivated by previous analysis of rigorous bounds on the permeability. Excellent agreement was found for a variety of model microstructures of porous media in the low and high porosity regimes. Experimentally obtained permeabilities of several sandstones were found to agree well with our universal curve.

We will review recent progress in our understanding of classical transport in porous media. Theoretical concepts will be illustrated with two distinct kinds of calculations. The first involve the grain consolidation model and are based on a particular multisize packing of spherical grains. The computational methods developed here are sufficiently accurate that we propose to combine them with direct measurements of the pore and grain geometry based on X-ray microtomography. Our preliminary results indicate that this approach may well play an important part in future studies of transport in porous media.

Recent progress that we have made on the problem of determining the effective properties of random heterogeneous media from the morphology is reviewed. A variety of different effective properties are considered, including the electrical conductivity and elastic moduli of composites, time scales associated with diffusion and reaction among traps, and fluid permeability of porous media. Studying seemingly disparate properties from a general viewpoint is shown to be fruitful.

Recent progress that we have made on the problem of determining the macroscopic properties of random heterogeneous media from the microstructure is reviewed. A variety of different effective properties are considered, including the effective thermal (or electrical) conductivity and elastic moduli of composites, fluid permeability of porous media, and effective time scales associated with diffusion and reaction among traps. A unified approach to treat these seemingly disparate problems is emphasized.

By using the most sensitive two-point correlation functions introduced to date, we reconstruct the microstructures of two-phase random media with heretofore unattained accuracy. Such media arise in a host of contexts, including porous and composite media, ecological structures, biological media, and astrophysical structures. The aforementioned correlation functions are special cases of the so-called {\it canonical} $n$-point correlation function $H_n$ and generalize the ones that have been recently employed to advance our ability to reconstruct complex microstructures [Y. Jiao, F. H. Stillinger, and S. Torquato, Proc. Nat. Acad. Sci. {\bf 106}, 17634 (2009)]. The use of these generalized correlation functions is tantamount to dilating or eroding a reference phase of the target medium and incorporating the additional topological information of the modified media, thereby providing more accurate reconstructions of percolating, filamentary, and other topologically complex microstructures. We apply our methods to a multiply-connected "donut" medium and a dilute distribution of "cracks" (a set of essentially zero measure) demonstrating improved accuracy in both cases with implications for higher-dimensional and biconnected two-phase systems. The high information content of the generalized two-point correlation functions suggests that it would be profitable to explore their use to characterize the structural and physical properties of not only random media, but also molecular systems, including structural glasses., Comment: 12 pages, 4 figures

Accepting the hypothesis that cancers are self-organizing, opportunistic systems, it is crucial to understand the collective behavior of cancer cells in their tumorous heterogeneous environment. In the present paper, we ask the following basic question: Is this self-organization of tumor evolution reflected in the manner in which malignant cells are spatially distributed in their heterogeneous environment? We employ a variety of nontrivial statistical microstructural descriptors that arise in the theory of heterogeneous media to characterize the spatial distributions of the nuclei of both benign brain white matter cells and brain glioma cells as obtained from histological images. These descriptors, which include the pair correlation function, structure factor and various nearest neighbor functions, quantify how pairs of cell nuclei are correlated in space in various ways. We map the centroids of the cell nuclei into point distributions to show that while commonly used local spatial statistics (e.g., cell areas and number of neighboring cells) cannot clearly distinguish spatial correlations in distributions of normal and abnormal cell nuclei, their salient structural features are captured very well by the aforementioned microstructural descriptors. We show that the tumorous cells pack more densely than normal cells and exhibit stronger effective repulsions between any pair of cells. Moreover, we demonstrate that brain gliomas are organized in a collective way rather than randomly on intermediate and large length scales. The existence of nontrivial spatial correlations between the abnormal cells strongly supports the view that cancer is not an unorganized collection of malignant cells but rather a complex emergent integrated system.